The big idea: A function is increasing on an interval when the graph goes up as you move from left to right.
It is decreasing when the graph goes down.
The key word is interval — we describe the range of x-values, not individual points.
- Increasing interval
- A range of x-values where f(x) gets larger as x increases. The graph rises left to right.
- Decreasing interval
- A range of x-values where f(x) gets smaller as x increases. The graph falls left to right.
- Stationary point
- A point where the graph is neither increasing nor decreasing — the gradient is zero.
Increasing (↗)
- Graph rises as you move left to right
- f(x) gets larger as x increases
- Happens before a local maximum and after a local minimum
- Express as: x < a or x > b
Decreasing (↘)
- Graph falls as you move left to right
- f(x) gets smaller as x increases
- Happens between a local maximum and the next minimum
- Express as: a < x < b
Intervals, not points: IB asks for intervals, not individual x-values.
Your answer must be a range such as 'x > 2' or '1 < x < 4'.
Saying 'x = 2' describes a single point — that is not an interval.
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The big idea: Find where the graph changes direction — those are the turning points.
The increasing and decreasing intervals are the x-ranges between (and beyond) the turning points.
Reading increasing/decreasing intervals
A graph has a local maximum at x = 1 and a local minimum at x = 4.
State where f is increasing and decreasing (assume f is defined for all x).
Step by step
- Increasing: graph rises left to right. This happens before x = 1 and after x = 4.
- Decreasing: graph falls left to right. This happens between the two turning points.
Final answer
Increasing for x < 1 and x > 4. Decreasing for 1 < x < 4.
Include or exclude the endpoints?: At a turning point the graph is momentarily flat — not increasing or decreasing.
IB accepts both strict (x < 1) and non-strict (x ≤ 1) inequalities here.
Use strict inequalities to be safe.
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The big idea: The two most common errors are: (1) giving a single x-value instead of an interval, and (2) mixing up which side is increasing vs decreasing.
Always look at the graph — does the curve rise or fall as you move right?
Wrong
- f is increasing at x = 2
- Increasing interval is y > 3
- Decreasing for x = 1 to x = 4 (no inequality)
Correct
- f is increasing for x < 2 (a range, not a point)
- Increasing/decreasing intervals use x-values, not y-values
- Decreasing for 1 < x < 4
IB question patterns to recognise
- 'State the interval(s) where f is increasing' — give x-range(s) using inequalities
- 'For what values of x is f decreasing?' — same format
- 'Describe the behaviour of f for x > 5' — state whether it is increasing, decreasing or approaching an asymptote
IB-style question — increasing/decreasing in context [4 marks]
A start-up's monthly profit, P thousand dollars, t months after launch is modelled by P(t) = −t² + 8t − 7, for 0 ≤ t ≤ 8.
(a) Find the month in which the profit is greatest.
(b) State the interval of months during which the profit is decreasing.
Step by step
- The profit curve is a downward parabola, so it rises to a single peak (the maximum) and then falls. Find the peak with the GDC, or from the vertex.
- (a) The greatest profit occurs at the turning point.
- (b) After the peak the profit falls. Decreasing runs from the maximum to the end of the domain.
- Give the decreasing interval as a range of t-values, and tie it back to the context — profit is falling from month 4 onward.
Final answer
(a) Month 4. (b) Profit is decreasing for 4 < t ≤ 8.
The big idea: When a function models a real situation, an increasing interval means the quantity is growing and a decreasing interval means it is shrinking.
Always interpret in the context of the problem.
Contextual interpretation
The temperature in a room T(t) is modelled over 10 hours.
The graph shows T increases for 0 < t < 3, then decreases for 3 < t < 10.
What does this tell us?
Step by step
- Increasing interval: 0 < t < 3 → temperature is rising for the first 3 hours.
- Decreasing interval: 3 < t < 10 → temperature is falling from hour 3 to hour 10.
- Turning point at t = 3 → the room was at its hottest after 3 hours.
Final answer
The room heats up for 3 hours then cools down for 7 hours. Maximum temperature occurs at t = 3.
State increasing/decreasing and interpret: IB context questions often ask you to do both: (1) identify the interval and (2) explain what it means.
Answer in two sentences: 'f is increasing for 0 < x < 3.
This means the temperature is rising during the first 3 hours.'